Optical diffuse reflection is a measurement technique used to probe the optical properties of turbid, i.e., scattering, samples. The technique is especially popular in the visible (VIS) and near-infrared (NIR) regions of the electromagnetic spectrum, where hardware is relatively cheap, and typical applications include industrial process control measurements, e.g., measurement of humidity in paper, and biomedical measurements, e.g., measurement of fat concentration in the skin. Major advantages of diffuse reflection over other optical sampling techniques include (a) the minimal need for sample preparation, because many practically important types of samples are scattering by nature, and (b) the possibility to perform measurements on bulky samples.
Most applications require spectroscopic measurements, i.e., diffuse reflection is measured at multiple optical wavelengths, λi(i=1,2 . . . ), and then so-called multivariate calibration is used to determine an algorithm that transforms the measured spectrum, S(λi) [Watt], into a single user-desired output number. The optimal algorithm to transform the multivariate input data, S(λi), into the desired analytical result is the so-called specific multivariate Wiener filter. (In practice, the unit of S(λ) varies with the instrument hardware and can also be, e.g., [Volt], [Ampere], [Coulomb], or [Digital Number], but all of these are proportional to detected optical power so without loss of generality, we will consistently use [Watt] here.) Data from multivariate measurements usually have much better accuracy than univariate data, because noise that is correlated between different inputs can be subtracted out in the algorithm. This statement has recently been cast into quantitative form by deriving the formula for the signal-to-noise ratio of a multivariate measurement, SNRX=s/neff, where s is the root-mean-square (rms) signal, and
                              n          eff                =                  1                                                    g                T                            ·                              N                x                                  -                  1                                            ·              g                                                          Eq        .                                  ⁢                  (          1          )                    is the rms effective noise of the multivariate data. Here, s and neff are measured in some user-defined and application-specific [concentration unit] of the property or analyte of interest, e.g., [gram/Liter]; Nx the covariance matrix of the multivariate noise in the measured data, in some [measurement unit squared], e.g., [Volt squared]; and g the so-called response vector, in [measurement unit/concentration unit]. For details, see R. Marbach, On Wiener filtering and the physics behind statistical modeling. Journal of Biomedical Optics 7, 130–147 (2002) and also, A New method for multivariate calibration, J. Near Infrared Spectrosc. 13, 241–254 (2005), where the latter paper is application-oriented and straightforward to read. High measurement accuracy is equivalent to low effective noise, neff, which, for a given rms signal s, is equivalent to a high SNRX. Equation (1) will repeatedly be referred to below, because it provides the motivation for much of the following. Commercially successful analytical measurements based on multivariate data generally achieve SNRX>3.
A disadvantage of the diffuse reflection technique is the lack of a simple formula that could provide an exactly linear relationship between the measured spectrum, S(λ), and the quantity of interest, e.g., concentration of a specific component in the sample. Loosely speaking, nothing as simple as the Lambert-Beer law is available. In practice, however, this is not a severe limitation because often, only small changes around a stable, average sample are observed and these small changes behave approximately linear. Linearity can be further improved by applying software pre-processing steps to the measured spectrum, S(λ), before feeding it into the calibration algorithm. The details of the pre-processing depend on the details of the application and used hardware and, to some extent, historical preferences. Various options are available and have been discussed in the pertinent literature. The most commonly used pre-processing steps include taking the logarithm, −log{S(λ)/Watt}, or when a spectrum SR(λi) from a reference standard is available, −log{S(λi)/SR(λi)}; and mean centering; see, e.g., Peter R. Griffiths, Letter: Practical consequences of math pre-treatment of near infrared reflectance data: log (1/R) vs F(R), J. Near Infrared Spectrosc. 3, 60–62 (1995). The popularity of taking the logarithm is also directly explained by Eq.(1): while not necessarily optimal at linearizing the spectral signal, the logarithm does perform a virtually perfect job at subtracting out spectral noises, and thus usually outperforms other, competing methods in the resulting signal-to-noise ratio,
                              SNR          X                =                  s          ⁢                                                                      g                  T                                ⁢                                  N                  x                                      -                    1                                                  ⁢                g                                      .                                              Eq        .                                  ⁢                  (          2          )                    
Most diffuse reflection measurements in use today are ratiometric measurements, i.e., the optical signal from the sample, S(λ), is ratioed by the optical signal received from a reflectance standard, SR(λ) [Watt], to produce the diffuse reflection spectrum, R(λ)=S(λ)/SR(λ), or the absorbance-equivalent, −log{S(λ)/SR(λ)}, after the most typical pre-processing. This ratio operation is known as “spectroscopic referencing” and instruments often contain a built-in reflectance standard that is moved in and out of the measurement position before or after the sample measurement. From the point of view of qualitative interpretation of the measured spectra, the purpose of referencing is to isolate the spectral signature of the sample by ratioing out the spectral characteristics of the instrument; thus, reflectance standards usually have as flat a reflection spectrum, SR(λ)≅const(λ), as possible over the wavelength range of interest. From the point of view of accuracy of a quantitative measurement, referencing decreases the spectral noise in the larger eigenfactors of the covariance matrix of the spectral noise, Nx, at the expense of a slight noise increase in the smaller eigenfactors (cmp. Marbach 2002). Since the spectral signal usually resides in the larger eigenfactors, referencing usually results in a net improvement of SNRX a.k.a. measurement accuracy.
Most diffuse reflection probes in use today are based on bifurcated fiber optic cables, in which illumination and collection fibers coming from two separate arms of the cable are combined and arranged in close proximity at the distal end of the cable (the probing end). In the VIS range, plastic fibers can be used, but glass fibers are used in most designs. All three ends of the cable are usually ground flat and polished after the fibers have been fixed into place. The total number of fibers used in typical probes varies, from two to several hundred, and various geometries with different relative merits exist and are in common use. One of the most important design parameters of a fiber probe is lateral distance, i.e., the (average) distance between the illumination fiber(s) and the collection fiber(s) measured along the surface of the sample. Lateral distance, along with the optical properties of the sample, largely determines the throughput efficiency of the probe (fraction of the light that is collected back) as well as the average penetration depth of the measurement light into the sample. One important advantage of fiber probes is the low sensitivity to surface (a.k.a. Fresnel) reflections at the sample surface. Surface reflections generally do not carry desired information about the sample and are usually detrimental to measurement accuracy. Non-fiber probes, which are based on conventional optical elements like windows, lenses, mirrors, beamsplitters, etc.; can achieve higher throughput efficiency than fiber probes but are generally more expensive and harder to use; for an example of a very high-throughput, non-fiber probe, see R. Marbach and H. M. Heise, Optical Diffuse Reflectance Accessory for Measurements of Skin Tissue by Near-Infrared Spectroscopy, Appl. Optics 34, 610–621 (1995). Because the areas of light entry into the sample, and light exit from the sample, are usually spatially-overlapping in non-fiber probes, surface reflections are usually a bigger problem than in fiber probes.
A dominant problem in many diffuse reflection measurements is instability generated by various physical effects located at or near the surface of the sample. Physical reasons include various microscopic, yet optically relevant, changes due to, e.g., dust, scratches in the glass of the probe, changes in the contact pressure between the probe and the sample, or slow variations of the optical properties of the surface layer of the sample due to changes in the ambient. We use the terms surface instabilities or surface noise to describe all of these effects, regardless of their physical origin. An example of an application suffering from surface instabilities is the diffuse reflection measurement of the scattering coefficient, μS, of human skin, which has been proposed as an indicator measurement closely correlated to changes in the blood glucose concentration. Experiments have shown that +1 mmol/Liter of change in blood glucose concentration causes a relatively large change of about ΔμS/μS≅−0.3% in in-vivo skin, see L. Heinemann et al., Noninvasive Glucose Measurement by Monitoring of scattering Coefficient During Oral Glucose Tolerance Tests, Diabetes Technology & Therapeutics 2, 211–220 (2000). Unfortunately, however, the instability of the measurement turned out to be much larger. Tests on healthy volunteers suffered from up to 5% of magnitude of drift over a measurement time of a few hours, in spite of the fact that the fiber probes were glued to the abdomen and volunteers were lying in bed as still as possible. From the time profiles of the drift, the authors concluded that accumulation of moisture underneath the probe due to the ever-present, unnoticeable slight sweating (“transepidermal water loss”) was a major reason for the observed drift. Sweating is a typical surface instability effect in many biomedical measurements.
The following paragraph will introduce notation used throughout the remainder of this document.
The vast majority of diffuse reflection probes in use today employ only a single ‘measurement beam’ through the sample, i.e., there is only a single ‘point of light entry’ into the sample and only a single ‘point of light exit.’ The phrase ‘point of entry’ here is used in a general sense meaning ‘sample surface area of light entrance of a measurement beam,’ and ditto for light exit. The definition of ‘measurement beam’ goes hand in hand with the definitions of entry and exit point and is as follows: a measurement beam is composed of measurement light that has probed the sample and is individually detected by the measurement instrument. For example, in the case of a simple fiber optic probe with only one illumination and one pickup fiber, both butted against the sample surface, the point of light entry is the distal face of the core of the illumination fiber and the point of light exit is the distal face of the core of the pickup fiber. Between the two and through the sample extends the one measurement beam. In the following, we will call this a 1×1 probe, meaning, 1 entry point ×1 exit point. Assuming the probe contained, say, 200 illumination fibers illuminated by one light source and 30 pickup fibers guiding light to one photodetector, then this would still be a 1×1 probe but now with the single point of light entry consisting of 200 circular areas and the single point of light exit consisting of 30 circular areas. However, if the 200 illumination fibers were divided into two groups of, say, 100 fibers each and illumination of the two groups was modulated and the electrical output of the single photodetector was demodulated such that the signals from the two groups were detected individually, e.g., by illuminating the two groups in alternating sequence, then there would be two points of light entry with 100 circles each and still one point of light exit with 30 circles. This last example is called a 2×1 probe and forms two measurement beams through the sample. As a theoretical maximum, a probe with 200 illumination fibers and 30 pickup fibers could establish 200 points of entry and 30 points of exit and could form a 200×30 probe, but that would require that 200×30=6000 measurement beams be individually modulated and detected, which is not feasible. In practice, multiple fibers are combined and the vast majority of probes today are 1×1 probes. Fibers are usually combined optically by forming fiber bundles, but sometimes, pickup fibers are combined electrically by adding the electrical signals from several photodetectors or are combined in software by effectively adding their signals after A/D conversion. In many cases, the one measurement beam of a 1×1 probe is spectroscopically broken down into multiple, individually detected signals from different wavelength bands, but in the discussed sense of beam geometry through the sample, these are still 1×1 probes. Lastly, it is not necessary that points of light entry and light exit are spatially separate from each other, but rather, they can overlap or even be identical, which is common in non-fiber probes.
A few probes have been built, mostly for research purposes, that use either multiple entry points or multiple exit points. The principal application here is the measurement of the so-called spatially resolved diffuse reflection spectrum, R(λ;r), where r[mm] is lateral distance. For example, the fiber probe used in the experiments described by Heinemann et al. 2000 consisted of a single illumination fiber and eight pickup fibers guided to form the input slit of an imaging spectrograph, thereby establishing a single point of entry and potentially eight points of exit (assuming that the eight signals had not been combined in software). As already stated above, however, the vast majority of diffuse reflection probes in use today are 1×1 probes, or are arrangements of multiple (independent) 1×1 probes dispersed over a spatially extended surface area of the sample for some type of imaging purpose.
A few more remarks concerning notation are necessary. First, we stress that the notation introduced above, viz., “Nin×Nout probe;” refers to more than just the number of light entry points (Nin) and light exit points (Nout). The notation also implies that measurement beams are established between all entry and exit points, i.e., that a total of (Nin×Nout)-many measurement beams are individually detected by the hardware. Next, in the following, two measurement beams are called signal-redundant whenever their measurement results are identical under the assumption of a spatially uniform sample. For example, in the hypothetical example of the 200×30 fiber probe above, many of the 6000 beams would be signal redundant because the geometry of their fibers and their lateral distance would be identical. Further, the term diffuse reflection is commonly used whenever the entry and exit points are located close to each other and on the same macroscopic side of the sample, as opposed to other terms like diffuse transmittance etc., which are commonly used when the two are located on opposite sides of the sample. Schmitt and Kumar, however, have pointed out that there is no clear-cut distinction between reflection and transmission whenever the sample is turbid, because every situation is a mixture of both, so to speak, see J. M. Schmitt and G. Kumar, Spectral Distortions in Near-Infrared Spectroscopy of Turbid Materials, Appl. Spectrosc. 50, 1066–1073 (1996). The authors point out that at small lateral distances, reflection dominates because an increase in the scatter coefficient, μS, [mm−1], of the sample will cause an increase in the detected power, S(λ) [Watt]; however, at large lateral distances the measurement will actually behave like a transmission set-up, because an increase in μS, will cause a decrease in S(λ). Here, we use the term diffuse reflection regardless of the direction and magnitude of the change in S(λ) caused by a change in μS, and regardless of whether or not the sample actually has a macroscopic, planar surface and how the points of entry and exit are oriented with respect to the sample. Also, we use the term diffuse reflection even if the measurement light, or part of it, is spatially coherent (e.g., from a superluminescent LED with single transverse mode) or time-coherent (e.g., from a laser with single longitudinal mode). In summary, we use the term diffuse reflection whenever a turbid sample is probed with an optical measurement beam that makes use of, or is subject to, scattering inside the sample.